X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=assignment3.mdwn;h=775eb79e571e78cfdca489121751b0262f9e4c07;hp=d00dba0b2cbbfa44a6adc84668e145aa6e96a7b6;hb=101b2a435c05ae480eaefaf30348e80bd2d3de5d;hpb=97cbee46cb35e1eb7aa5ea5f8b1af3583f5b7522 diff --git a/assignment3.mdwn b/assignment3.mdwn index d00dba0b..775eb79e 100644 --- a/assignment3.mdwn +++ b/assignment3.mdwn @@ -6,90 +6,127 @@ assignment much faster and more secure. *Writing recursive functions on version 1 style lists* -Recall that version 1 style lists are constructed like this: +Recall that version 1 style lists are constructed like this (see +[[lists and numbers]]): - - -do eta-reductions too - - - - - -
```+eq 2 2 yes no
```
- + + +Then `length mylist` evaluates to 3. + +1. What does `head (tail (tail mylist))` evaluate to? + +2. Using the `length` function as a model, and using the predecessor +function, write a function that computes factorials. (Recall that n!, +the factorial of n, is n times the factorial of n-1.) + +Warning: my browser isn't able to compute factorials of numbers +greater than 2 (it does't provide enough resources for the JavaScript +interpreter; web pages are not supposed to be that computationally +intensive). + +3. (Easy) Write a function `listLenEq` that returns true just in case two lists have the +same length. That is, + + listLenEq mylist (makeList meh (makeList meh (makeList meh nil))) ~~> true + + listLenEq mylist (makeList meh (makeList meh nil))) ~~> false + + +4. (Still easy) Now write the same function, but don't use the length function. + +5. In assignment 2, we discovered that version 3-type lists (the ones that +work like Church numerals) made it much easier to define operations +like `map` and `filter`. But now that we have recursion in our toolbox, +reasonable map and filter functions for version 3 lists are within our +reach. Give definitions for `map` and a `filter` for verson 1 type lists. + +6. Linguists analyze natural language expressions into trees. +We'll need trees in future weeks, and tree structures provide good +opportunities for learning how to write recursive functions. +Making use of the resources we have at the moment, we can approximate +trees as follows: instead of words, we'll use Church numerals. +Then a tree will be a (version 1 type) list in which each element is +itself a tree. For simplicity, we'll adopt the convention that +a tree of length 1 must contain a number as its only element. +Then we have the following representations: + +
```+   (a)           (b)             (c)
+    .
+   /|\            /\              /\
+  / | \          /\ 3             1/\
+  1 2  3        1  2               2 3
+
+[[1];[2];[3]]  [[[1];[2]];[3]]   [[1];[[2];[3]]]
+```
+ +Limitations of this scheme include the following: there is no easy way +to label a constituent with a syntactic category (S or NP or VP, +etc.), and there is no way to represent a tree in which a mother has a +single daughter. + +When processing a tree, you can test for whether the tree contains +only a numeral (in which case the tree is leaf node) by testing for +whether the length of the list is less than or equal to 1. This will +be your base case for your recursive functions that operate on these +trees. + +#Write a function that sums the number of leaves in a tree.# +Expected behavior: + +
```+
+let t1 = (make-list 1 nil) in
+let t2 = (make-list 2 nil) in
+let t3 = (make-list 3 nil) in
+let t12 = (make-list t1 (make-list t2 nil)) in
+let t23 = (make-list t2 (make-list t3 nil)) in
+let ta = (make-list t1 t23) in
+let tb = (make-list t12 t3) in
+let tc = (make-list t1 (make-list t23 nil)) in
+
+count-leaves t1 ~~> 1
+count-leaves t2 ~~> 2
+count-leaves t3 ~~> 3
+count-leaves t12 ~~> 3
+count-leaves t23 ~~> 5
+count-leaves ta ~~> 6
+count-leaves tb ~~> 6
+count-leaves tc ~~> 6
++
+#Write a function that counts the number of leaves.#
+
```